A function is a mathematical relation that connects inputs to outputs. The key rule is that each input must correspond to exactly one output. Here we see a function f(x) equals 2x, where input 1 maps to output 2, input 2 maps to output 4, and input 3 maps to output 6.
Functions use special notation. We write f of x to represent a function named f with input x. For example, f(x) equals x plus 3 means the function adds 3 to any input. Similarly, g(x) equals x squared, and h(x) equals 2x minus 1. Each function has its own unique graph showing how inputs relate to outputs.
Every function has a domain and range. The domain is the set of all possible input values, while the range is the set of all possible output values. For the function f(x) equals x squared, the domain includes all real numbers since we can square any number. However, the range only includes values greater than or equal to zero, because squaring any real number always gives a non-negative result.
To determine if a graph represents a function, we use the vertical line test. If any vertical line crosses the graph at most once, it's a function. If a vertical line crosses more than once, it's not a function. The parabola y equals x squared passes this test because each vertical line intersects it only once. However, a circle fails the test because most vertical lines intersect it twice.
There are many types of functions, each with unique properties. Linear functions create straight lines, quadratic functions form parabolas, exponential functions show rapid growth or decay, and trigonometric functions create wave patterns. Functions are fundamental tools used across many fields including physics, engineering, economics, computer science, and medicine to model real-world relationships and solve complex problems.